A finiteness theorem for zero-cycles over \(p\)-adic fields
type: Item
instance of: scholarly article
title: A finiteness theorem for zero-cycles over \(p\)-adic fields
zbMATH Open document ID: 1210.14012
author: Shuji Saito
author: Kanetomo Sato
published in: Annals of Mathematics. Second Series
publication date: 2011-02-11T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0605165
review text: Let \(V\) be a smooth projective variety over a \(p\)-adic field \(k\). Let \(CH_0(V)\) and \(A_0(V)\) be the group of zero-cycles on \(V\) modulo rational equivalence and its subgroup consisting of cycles of degree zero. It is conjectured that \(A_0(V)\) is the direct sum of a finite group and a \(p'\)-divisible group. (An abelian group is called \(p'\)-divisible iff it is divisible by any integer prime to \(p\).) In the paper under review, this conjecture is proved under a weak assumption, namely that \(V\) has a regular projective flat model over the ring of integers in \(k\) on which the reduced subscheme of the special fiber has simple normal crossings. This result has not been known even when \(V\) has good reduction. This result is deduced from the following strong theorem on a cycle map, which is proved in wider context. Let \(R\) be an excellent henselian discrete valuation ring with residue field \(F\) and fraction field \(k\). (No assumption is made for the characteristic of \(k\).) Let \(X\) be a regular projective flat scheme over \(R\) on which the reduced subscheme of the special fiber has simple normal crossings. We write \(CH_1(X)\) for the Chow group of one-dimensional cycles on \(X\). Let \(n\) be a natural number prime to the characteristic of \(F\). There is a cycle class map \(\rho_X : CH_1(X)/n \to H^{2d}_{et}(X, \mathbb{Z}/n\mathbb{Z}(d))\), where \(d = \dim X - 1\). The following strong theorem is proved in this paper: If \(F\) is finite or separably closed, then \(\rho_X\) is bijective. This theorem is studied in the framework of Kato homology. We keep the same notation and assumption as the preceding paragraph. The Kato homology \(KH_a(X, \mathbb{Z}/n\mathbb{Z})\) is defined to be the homology group of a complex of the form \[ \bigoplus_{x \in X_0} H^0_{et}(x, \mathbb{Z}/n\mathbb{Z}(-1)) \leftarrow \cdots \leftarrow \bigoplus_{x \in X_a} H^a_{et}(x, \mathbb{Z}/n\mathbb{Z}(a-1)) \leftarrow \cdots \] in degree \(a\). (The leftmost term is pl…
reviewed by: Takao Yamazaki
Mathematics Subject Classification ID: 14C15
Mathematics Subject Classification ID: 14G20
zbMATH DE Number: 5850183
zbMATH Keywords: Chow group
zbMATH Keywords: cycle map
zbMATH Keywords: Kato homology
zbMATH Keywords: affine Lefschetz
zbMATH Keywords: Bertini theorem
MaRDI profile type: Publication
OpenAlex ID: W2135662680
arXiv ID: math/0605165
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DOI: 10.4007/ANNALS.2010.172.1593
description: scientific article